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Consider the Rock, Paper, Scissors game. Now we have three players. The rules are the same as two-player game: Rock smashes Scissors (rock wins); Scissors cuts Paper (scissors win); Paper covers Rock (paper wins). It is still a zero-sum game. But now the player could win 2 times of bet if he wins both of his opponents. Also, he could win one of his opponents and lose to his the other opponent. Or he may lose to both of his opponents.

Similarly as the two-player game, there does not exist a pure strategic equilibrium. By the existence of Nash equilibrium in finite-player game with finitely many strategies, there exists at least a mixed strategy Nash equilibrium.

For the completely mixed strategy Nash equilibrium, consider the strategy profile that every player chooses Rock, Paper, Scissors with equal probability (1/3, 1/3, 1/3). You can prove this is indeed a completely mixed strategy Nash equilibrium.[Manage messages]

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I am trying to design a payoff scheme for 3-player, 3-action static game such that none of the probabilities of any mixed strategy equilibrium is zero. In other words, the support for all 3 players should include all 3 actions [View full text and thread]